How can I set index while converting dictionary to dataframe?

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Use set_index:

df = pd.DataFrame(dictionary, columns=['Date', 'Open', 'Close'])  
df = df.set_index('Date')       
print (df)
                      Open  Close
Date                             
2016/11/22 07:00:00  47.47  47.48
2016/11/22 06:59:00  47.46  47.45
2016/11/22 06:58:00  47.38  47.40

Or use inplace:

df = pd.DataFrame(dictionary, columns=['Date', 'Open', 'Close'])  
df.set_index('Date', inplace=True)       
print (df)
                      Open  Close
Date                             
2016/11/22 07:00:00  47.47  47.48
2016/11/22 06:59:00  47.46  47.45
2016/11/22 06:58:00  47.38  47.40

Another possible solution filter out dict by Date key and then set index by dictionary['Date']:

df = pd.DataFrame({k: v for k, v in dictionary.items() if not k == 'Date'}, 
                   index=dictionary['Date'], 
                   columns=['Open','Close'])  
df.index.name="Date"
print (df)
                      Open  Close
Date                             
2016/11/22 07:00:00  47.47  47.48
2016/11/22 06:59:00  47.46  47.45
2016/11/22 06:58:00  47.38  47.40

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